The one-dimensional wave equation with general boundary conditions

Abstract

We show that a realization of the Laplace operator Au := u′′ with general nonlocal Robin boundary conditions α _j u′(j) + β _j u(j) + γ _1–j u(1 − j) = 0, (j = 0, 1) generates a cosine family on L ^p(0, 1) for every ${p\,{\in}\,[1,\infty)}$. Here α _j , β _j and γ _j are complex numbers satisfying α ₀, α ₁ ≠ 0. We also obtain an explicit representation of local solutions to the associated wave equation by using the classical d’Alembert’s formula.